= Cartan curvature forms for a static spherical metric
{c}
{title2=$g=-Vdt^2+V^{-1}dr^2+r^2d\Omega^2$}
In the exterior orthonormal coframe $E^0=\sqrt Vdt$, $E^1=dr/\sqrt V$, $E^2=rd\theta$, $E^3=r\sin\theta d\phi$, use $\Omega^a{}_b=\tfrac12R^a{}_{bcd}E^c\wedge E^d$. The independent mixed-index <curvature 2-forms> are $\Omega^0{}_1=-V''E^0\wedge E^1/2$, $\Omega^0{}_A=-V'E^0\wedge E^A/(2r)$, $\Omega^1{}_A=-V'E^1\wedge E^A/(2r)$ for $A=2,3$, and $\Omega^2{}_3=(1-V)E^2\wedge E^3/r^2$. The orthonormal <Ricci tensor> is diagonal with entries $V''/2+V'/r$, $-V''/2-V'/r$, and twice $(1-V-rV')/r^2$. The <Einstein field equations> with negative <cosmological constant> then give $V=1-2M/r+r^2/l^2$.
// Target: general-relativity.bigb
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